Ombrian curves (from the Greek ombros, meaning rainfall) are most widely known as
rainfall intensity-duration-frequency (IDF) curves or relationships. However, the former
term may be preferable as the later is inaccurate. Namely, "frequency" is meant
to be "return period" where as "duration" is in fact the "time scale" on which the rainfall
process is averaged. Thus, ombrian relationships are nothing more than multiple
time scale expressions of the rainfall probability. Three important issues regarding the
mathematical form of the ombrian relationships are examined: (a) whether or not the
effects of time scale and return period are separable so that the relationship could be
written as the product of two scalar functions; (b) whether or not the rainfall intensity
is a power function of return period and (c) whether or not the rainfall is a power function
of time scale. All these questions are investigated using the principle of maximum
entropy as a theoretical basis and a long rainfall data set as an empirical basis. It turns
out that none of the above questions has a precisely positive answer, which makes
the theoretical derivation of ombrian curves a complicated task. For this reason, consistent
approximations are sought, which eventually do not depart significantly from
commonly used forms in engineering practice.